Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.975 - 0.219i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.243 + 0.907i)2-s + (−1.31 − 1.12i)3-s + (0.967 − 0.558i)4-s + (1.66 + 1.49i)5-s + (0.700 − 1.46i)6-s + (0.0144 − 2.64i)7-s + (2.07 + 2.07i)8-s + (0.470 + 2.96i)9-s + (−0.949 + 1.87i)10-s + (0.630 − 0.363i)11-s + (−1.90 − 0.352i)12-s + (−1.44 + 1.44i)13-s + (2.40 − 0.630i)14-s + (−0.515 − 3.83i)15-s + (−0.257 + 0.446i)16-s + (−7.09 − 1.90i)17-s + ⋯
L(s)  = 1  + (0.171 + 0.641i)2-s + (−0.760 − 0.649i)3-s + (0.483 − 0.279i)4-s + (0.744 + 0.667i)5-s + (0.285 − 0.599i)6-s + (0.00544 − 0.999i)7-s + (0.732 + 0.732i)8-s + (0.156 + 0.987i)9-s + (−0.300 + 0.592i)10-s + (0.189 − 0.109i)11-s + (−0.549 − 0.101i)12-s + (−0.400 + 0.400i)13-s + (0.642 − 0.168i)14-s + (−0.133 − 0.991i)15-s + (−0.0644 + 0.111i)16-s + (−1.71 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.975 - 0.219i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.975 - 0.219i)$
$L(1)$  $\approx$  $1.09887 + 0.122113i$
$L(\frac12)$  $\approx$  $1.09887 + 0.122113i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1.31 + 1.12i)T \)
5 \( 1 + (-1.66 - 1.49i)T \)
7 \( 1 + (-0.0144 + 2.64i)T \)
good2 \( 1 + (-0.243 - 0.907i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.630 + 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 - 1.44i)T - 13iT^{2} \)
17 \( 1 + (7.09 + 1.90i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.664 + 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 - 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.08 + 1.63i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (-1.82 - 6.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.41 + 5.26i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.807 + 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.84 + 6.90i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.06iT - 71T^{2} \)
73 \( 1 + (-15.2 - 4.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.80 + 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.83 + 1.83i)T + 83iT^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.82644195498246144195401129205, −13.10740430401115278599030400520, −11.36585604368992257682223897194, −10.94645698602392736905813221196, −9.750815328162047964493103269303, −7.71457861236073919943853818382, −6.78560622612224371978551636261, −6.22553101429231411132852544659, −4.77478235978827560227104249067, −2.04814462909678929295636779610, 2.22442289446052811584306463215, 4.19793706280072383652498011781, 5.53383447784735967982902130991, 6.63830762211975363097745162227, 8.620144448583061485018753819444, 9.694009145111298554574264829329, 10.68305007261624622088859968040, 11.71759249037398741037596282571, 12.46732676408580339138999508366, 13.27953235084798891518565764900

Graph of the $Z$-function along the critical line